AP®MathsCollege BoardCalculus BCRevision NotesUnit 10: Infinite Sequences and SeriesTests for Divergence & ConvergenceAlternating Series Test for Convergence

Alternating Series Test for Convergence (College Board AP® Calculus BC): Revision Note

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 19 May 2026

Alternating series test

What is an alternating series?

An alternating series is a series where the terms alternate between negative and positive
- For example the alternating harmonic series $\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n + 1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + . . .$
Alternating series can be written in one of the following two forms, where in each case $a_{n} > 0$ for each value of $n$
- $\sum_{n = 1}^{\infty} {(- 1)}^{n + 1} \cdot a_{n} = a_{1} - a_{2} + a_{3} - a_{4} + . . .$
  - This version starts with a positive term
- $\sum_{n = 1}^{\infty} {(- 1)}^{n} \cdot a_{n} = - a_{1} + a_{2} - a_{3} + a_{4} - . . .$
  - This version starts with a negative term

What is the alternating series test for convergence?

The alternating series test is a method for determining whether an alternating series converges
The alternating series test states that:
- Given an alternating series of the form $\sum_{n = 1}^{\infty} {(- 1)}^{n + 1} \cdot a_{n}$ or $\sum_{n = 1}^{\infty} {(- 1)}^{n} \cdot a_{n}$ with $a_{n} > 0$ for each value of $n$
- The series converges if
  - $a_{1} \geq a_{2} \geq a_{3} \geq . . . \geq a_{n} \geq . . .$
  - and $\lim_{n \to \infty} a_{n} = 0$
Note that you can't use the alternating series test to show that an alternating series diverges
- There are convergent alternating series for which $a_{1} \geq a_{2} \geq a_{3} \geq . . . \geq a_{n} \geq . . .$ is not true
- However $\lim_{n \to \infty} a_{n} = 0$ must be true for any series to converge (see the 'nth Term Test for Divergence' study guide)

Worked Example

Use the alternating series test to show that the alternating harmonic series $\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n + 1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + . . .$ converges.

Answer:

The harmonic series can be rewritten as $\sum_{n = 1}^{\infty} {(- 1)}^{n + 1} \cdot \frac{1}{n}$ , i.e. in 'standard' alternating series form with $a_{n} = \frac{1}{n}$

First show that $a_{1} \geq a_{2} \geq a_{3} \geq . . . \geq a_{n} \geq . . .$ is true

$1 > \frac{1}{2} > \frac{1}{3} > \frac{1}{4} > . . . > \frac{1}{n} > . . .$

Now check the $\lim_{n \to \infty} a_{n} = 0$ condition

$\lim_{n \to \infty} \frac{1}{n} = 0$

Therefore, both conditions of the alternating series test have been met

By the alternating series test, $\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n + 1}}{n}$ converges

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